Framework to Study Dynamic Dependencies in Networks of Interacting Processes

The analysis of dynamic dependencies in complex systems such as the brain helps to understand how emerging properties arise from interactions. Here we propose an information-theoretic framework to analyze the dynamic dependencies in multivariate time-evolving systems. This framework constitutes a fully multivariate extension and unification of previous approaches based on bivariate or conditional mutual information and Granger causality or transfer entropy. We define multi-information measures that allow us to study the global statistical structure of the system as a whole, the total dependence between subsystems, and the temporal statistical structure of each subsystem. We develop a stationary and a nonstationary formulation of the framework. We then examine different decompositions of these multi-information measures. The transfer entropy naturally appears as a term in some of these decompositions. This allows us to examine its properties not as an isolated measure of interdependence but in the context of the complete framework. More generally we use causal graphs to study the specificity and sensitivity of all the measures appearing in these decompositions to different sources of statistical dependence arising from the causal connections between the subsystems. We illustrate that there is no straightforward relation between the strength of specific connections and specific terms in the decompositions. Furthermore, causal and noncausal statistical dependencies are not separable. In particular, the transfer entropy can be nonmonotonic in dependence on the connectivity strength between subsystems and is also sensitive to internal changes of the subsystems, so it should not be interpreted as a measure of connectivity strength. Altogether, in comparison to an analysis based on single isolated measures of interdependence, this framework is more powerful to analyze emergent properties in multivariate systems and to characterize functionally relevant changes in the dynamics.